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<h1 class="title">国赛2024A题 “板凳龙” 闹元宵</h1>
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<section id="a-题-板凳龙-闹元宵" class="level1" data-number="1">
<h1 data-number="1"><span class="header-section-number">1</span> A 题 “板凳龙” 闹元宵</h1>
<p>“板凳龙”，又称“盘龙”，是浙闽地区的传统地方民俗文化活动。人们将少则几十条，多则上百条的板凳首尾相连，形成蜿蜒曲折的板凳龙。盘龙时，龙头在前领头，龙身和龙尾相随盘旋，整体呈圆盘状。一般来说，在舞龙队能够自如地盘入和盘出的前提下，盘龙所需要的面积越小、行进速度越快，则观赏性越好。</p>
<p>某板凳龙由 223 节板凳组成，其中第 1 节为龙头，后面 221 节为龙身，最后 1 节为龙尾。龙头的板长为 341 cm，龙身和龙尾的板长均为 220 cm，所有板凳的板宽均为 30 cm。每节板凳上均有两个孔，孔径（孔的直径）为 5.5 cm，孔的中心距离最近的板头 27.5 cm（见图 1 和图 2）。相邻两条板凳通过把手连接（见图 3）。</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="img/atu1.png" class="img-fluid figure-img"></p>
<figcaption>图1 龙头的俯视图</figcaption>
</figure>
</div>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="img/atu2.png" class="img-fluid figure-img"></p>
<figcaption>图2 龙身和龙尾的俯视图</figcaption>
</figure>
</div>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="img/atu3.png" class="img-fluid figure-img"></p>
<figcaption>图3 板凳的正视图</figcaption>
</figure>
</div>
<p>请建立数学模型，解决以下问题：</p>
<p><strong>问题 1</strong> 舞龙队沿螺距为 55 cm 的等距螺线顺时针盘入，各把手中心均位于螺线上。龙头前把手的行进速度始终保持 1 m/s。初始时，龙头位于螺线第 16 圈 A 点处（见图 4）。请给出从初始时刻到 300 s 为止，每秒整个舞龙队的位置和速度（指龙头、龙身和龙尾各前把手及龙尾后把手中心的位置和速度，下同），将结果保存到文件 result1.xlsx 中（模板文件见附件，其中“龙尾（后）”表示龙尾后把手，其余的均是前把手，结果保留 6 位小数，下同）。同时在论文中给出 0 s、60 s、120 s、180 s、240 s、300 s 时，龙头前把手、龙头后面第 1、51、101、151、201 节龙身前把手和龙尾后把手的位置和速度（格式见表 1 和表 2）。</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="img/atu4.png" class="img-fluid figure-img"></p>
<figcaption>图4 盘入螺线示意图</figcaption>
</figure>
</div>
<p>表 1 论文中位置结果的格式</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th style="text-align: center;"></th>
<th>0 s</th>
<th>60 s</th>
<th>120 s</th>
<th>180 s</th>
<th>240 s</th>
<th>300 s</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;">龙头 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">龙头 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 1 节龙身 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 1 节龙身 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 51 节龙身 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 51 节龙身 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 101 节龙身 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 101 节龙身 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 151 节龙身 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 151 节龙身 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 201 节龙身 x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 201 节龙身 y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">龙尾（后）x (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">龙尾（后）y (m)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p>表 2 论文中速度结果的格式</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th style="text-align: center;"></th>
<th>0 s</th>
<th>60 s</th>
<th>120 s</th>
<th>180 s</th>
<th>240 s</th>
<th>300 s</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;">龙头 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 1 节龙身 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 51 节龙身 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 101 节龙身 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">第 151 节龙身 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td style="text-align: center;">第 201 节龙身 (m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td style="text-align: center;">龙尾（后）(m/s)</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p><strong>问题 2</strong> 舞龙队沿问题 1 设定的螺线盘入，请确定舞龙队盘入的终止时刻，使得板凳之间不发生碰撞（即舞龙队不能再继续盘入的时间），并给出此时舞龙队的位置和速度，将结果存放到文件 result2.xlsx 中（模板文件见附件）。同时在论文中给出此时龙头前把手、龙头后面第 1、51、101、151、201 条龙身前把手和龙尾后把手的位置和速度。</p>
<p><strong>问题 3</strong> 从盘入到盘出，舞龙队将由顺时针盘入调头切换为逆时针盘出，这需要一定的调头空间。若调头空间是以螺线中心为圆心、直径为 9 m 的圆形区域（见图 5），请确定最小螺距，使得龙头前把手能够沿着相应的螺线盘入到调头空间的边界。</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="img/atu5.png" class="img-fluid figure-img"></p>
<figcaption>图5 调头空间示意图</figcaption>
</figure>
</div>
<p><strong>问题 4</strong> 盘入螺线的螺距为 1.7 m，盘出螺线与盘入螺线关于螺线中心呈中心对称，舞龙队在问题 3 设定的调头空间内完成调头，调头路径是由两段圆弧相切连接而成的 S 形曲线，前一段圆弧的半径是后一段的 2 倍，它与盘入、盘出螺线均相切。能否调整圆弧，仍保持各部分相切，使得调头曲线变短？</p>
<p>龙头前把手的行进速度始终保持 1 m/s。以调头开始时间为零时刻，给出从−100 s 开始到100 s 为止，每秒舞龙队的位置和速度，将结果存放到文件 result4.xlsx 中（模板文件见附件）。同时在论文中给出−100 s、−50 s、0 s、50 s、100 s 时，龙头前把手、龙头后面第 1、51、101、151、201 节龙身前把手和龙尾后把手的位置和速度。</p>
<p><strong>问题 5</strong> 舞龙队沿问题 4 设定的路径行进，龙头行进速度保持不变，请确定龙头的最大行进速度，使得舞龙队各把手的速度均不超过 2 m/s。</p>
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        // This is a special case and we should probably do some content thinning / targeting
        fetch(url)
        .then(res => res.text())
        .then(html => {
          const parser = new DOMParser();
          const htmlDoc = parser.parseFromString(html, "text/html");
          const note = htmlDoc.querySelector('main.content');
          if (note !== null) {
            // This should only happen for chapter cross references
            // (since there is no id in the URL)
            // remove the first header
            if (note.children.length > 0 && note.children[0].tagName === "HEADER") {
              note.children[0].remove();
            }
            const html = processXRef(null, note);
            instance.setContent(html);
          } 
        }).finally(() => {
          instance.enable();
          instance.show();
        });
      }
    }, function(instance) {
    });
  }
      let selectedAnnoteEl;
      const selectorForAnnotation = ( cell, annotation) => {
        let cellAttr = 'data-code-cell="' + cell + '"';
        let lineAttr = 'data-code-annotation="' +  annotation + '"';
        const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
        return selector;
      }
      const selectCodeLines = (annoteEl) => {
        const doc = window.document;
        const targetCell = annoteEl.getAttribute("data-target-cell");
        const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
        const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
        const lines = annoteSpan.getAttribute("data-code-lines").split(",");
        const lineIds = lines.map((line) => {
          return targetCell + "-" + line;
        })
        let top = null;
        let height = null;
        let parent = null;
        if (lineIds.length > 0) {
            //compute the position of the single el (top and bottom and make a div)
            const el = window.document.getElementById(lineIds[0]);
            top = el.offsetTop;
            height = el.offsetHeight;
            parent = el.parentElement.parentElement;
          if (lineIds.length > 1) {
            const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
            const bottom = lastEl.offsetTop + lastEl.offsetHeight;
            height = bottom - top;
          }
          if (top !== null && height !== null && parent !== null) {
            // cook up a div (if necessary) and position it 
            let div = window.document.getElementById("code-annotation-line-highlight");
            if (div === null) {
              div = window.document.createElement("div");
              div.setAttribute("id", "code-annotation-line-highlight");
              div.style.position = 'absolute';
              parent.appendChild(div);
            }
            div.style.top = top - 2 + "px";
            div.style.height = height + 4 + "px";
            div.style.left = 0;
            let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
            if (gutterDiv === null) {
              gutterDiv = window.document.createElement("div");
              gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
              gutterDiv.style.position = 'absolute';
              const codeCell = window.document.getElementById(targetCell);
              const gutter = codeCell.querySelector('.code-annotation-gutter');
              gutter.appendChild(gutterDiv);
            }
            gutterDiv.style.top = top - 2 + "px";
            gutterDiv.style.height = height + 4 + "px";
          }
          selectedAnnoteEl = annoteEl;
        }
      };
      const unselectCodeLines = () => {
        const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
        elementsIds.forEach((elId) => {
          const div = window.document.getElementById(elId);
          if (div) {
            div.remove();
          }
        });
        selectedAnnoteEl = undefined;
      };
        // Handle positioning of the toggle
    window.addEventListener(
      "resize",
      throttle(() => {
        elRect = undefined;
        if (selectedAnnoteEl) {
          selectCodeLines(selectedAnnoteEl);
        }
      }, 10)
    );
    function throttle(fn, ms) {
    let throttle = false;
    let timer;
      return (...args) => {
        if(!throttle) { // first call gets through
            fn.apply(this, args);
            throttle = true;
        } else { // all the others get throttled
            if(timer) clearTimeout(timer); // cancel #2
            timer = setTimeout(() => {
              fn.apply(this, args);
              timer = throttle = false;
            }, ms);
        }
      };
    }
      // Attach click handler to the DT
      const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
      for (const annoteDlNode of annoteDls) {
        annoteDlNode.addEventListener('click', (event) => {
          const clickedEl = event.target;
          if (clickedEl !== selectedAnnoteEl) {
            unselectCodeLines();
            const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
            if (activeEl) {
              activeEl.classList.remove('code-annotation-active');
            }
            selectCodeLines(clickedEl);
            clickedEl.classList.add('code-annotation-active');
          } else {
            // Unselect the line
            unselectCodeLines();
            clickedEl.classList.remove('code-annotation-active');
          }
        });
      }
  const findCites = (el) => {
    const parentEl = el.parentElement;
    if (parentEl) {
      const cites = parentEl.dataset.cites;
      if (cites) {
        return {
          el,
          cites: cites.split(' ')
        };
      } else {
        return findCites(el.parentElement)
      }
    } else {
      return undefined;
    }
  };
  var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
  for (var i=0; i<bibliorefs.length; i++) {
    const ref = bibliorefs[i];
    const citeInfo = findCites(ref);
    if (citeInfo) {
      tippyHover(citeInfo.el, function() {
        var popup = window.document.createElement('div');
        citeInfo.cites.forEach(function(cite) {
          var citeDiv = window.document.createElement('div');
          citeDiv.classList.add('hanging-indent');
          citeDiv.classList.add('csl-entry');
          var biblioDiv = window.document.getElementById('ref-' + cite);
          if (biblioDiv) {
            citeDiv.innerHTML = biblioDiv.innerHTML;
          }
          popup.appendChild(citeDiv);
        });
        return popup.innerHTML;
      });
    }
  }
});
</script>
</div> <!-- /content -->




</body></html>